4. Prove that if N = ak -. 2o in its decimal presentation then N = Lk-(-1)'ai (mod 11). Use this to find whether 13243546576879 is divisible by 11. 5(*) We have 1001 = 7.11. 13. Use this to prove that if N = ak... ao in its decimal presentation then N = 22212o a5a4a3 + aga726 - ... (mod 7) = 2221Q0 2504a3 + aga726 - ... (mod 11) = 22212o a5a4a3 + aga706 (mod 13). For example, 519285364 = 364 285 + 519 (mod 7) = 14 5+ 29 (mod 7) = 3 (mod 7), 519285364 = 364 285 + 519 (mod 11) =1+1+13 (mod 11) = 4 (mod 11) and519285364 = 364 285 + 519 (mod 13) = 0 25 1 (mod 13) = 0 (mod 13). This provides a criterion for divisibility by 7 although, admittedly, not a terribly convenient one. It allows to reduce the study of divisibility by 7 to that of 3 digit numbers. Now, since 10 = 3 (mod 7), 100 = 102 = 32 (mod 7) = 2 (mod 7), we have a2a1a0 = 20 + 3a1 + 2a2 (mod 7), and so for an arbitrary number we get an .ao = (ao - a3 + 26 - ...) + 3(a1 - 24+ a7 - ...) + 2(a2 as + ag - ...) (mod 7). Going back to our example, we can write 519285364 = 4 5+9+36 8+1) + 2(3 2+5) (mod 7) = 83 2 (mod 7) = 3 (mod 7). 4. Prove that if N = ak -. 2o in its decimal presentation then N = Lk-(-1)'ai (mod 11). Use this to find whether 13243546576879 is divisible by 11. 5(*) We have 1001 = 7.11. 13. Use this to prove that if N = ak... ao in its decimal presentation then N = 22212o a5a4a3 + aga726 - ... (mod 7) = 2221Q0 2504a3 + aga726 - ... (mod 11) = 22212o a5a4a3 + aga706 (mod 13). For example, 519285364 = 364 285 + 519 (mod 7) = 14 5+ 29 (mod 7) = 3 (mod 7), 519285364 = 364 285 + 519 (mod 11) =1+1+13 (mod 11) = 4 (mod 11) and519285364 = 364 285 + 519 (mod 13) = 0 25 1 (mod 13) = 0 (mod 13). This provides a criterion for divisibility by 7 although, admittedly, not a terribly convenient one. It allows to reduce the study of divisibility by 7 to that of 3 digit numbers. Now, since 10 = 3 (mod 7), 100 = 102 = 32 (mod 7) = 2 (mod 7), we have a2a1a0 = 20 + 3a1 + 2a2 (mod 7), and so for an arbitrary number we get an .ao = (ao - a3 + 26 - ...) + 3(a1 - 24+ a7 - ...) + 2(a2 as + ag - ...) (mod 7). Going back to our example, we can write 519285364 = 4 5+9+36 8+1) + 2(3 2+5) (mod 7) = 83 2 (mod 7) = 3 (mod 7)